We present an approach for merging boundary element method (BEM) solutions of the electric field in electrically insulating complex geometries with Brownian dynamics (BD) simulations of DNA electrophoresis therein. Although a rote application of the standard BEM algorithm proves inaccurate and prohibitively expensive, we show that regularization of the near-wall electric field and an updating scheme commensurate with the characteristic length scale of the BD simulation furnishes a robust, efficient simulation protocol. The accuracy of the BEM-BD method is verified by simulating lambda-DNA collisions with an isolated, insulating cylindrical obstacle and comparing the results with equivalent BD simulations that employ the exact solution for the electric field. The computational overhead of our implementation of BEM-BD is comparable to an existing finite element method/BD approach. The BEM-BD algorithm is readily parallelized and well-suited to time-dependent and responsive electric fields, making it broadly applicable to simulating DNA electrophoresis in microfluidic devices.